Model selection for regression on a random design

We consider the problem of estimating an unknown regression function
when the design is random with values in . Our estimation
procedure is based on model selection and does not rely on any prior
information on the target function. We start with a collection of
linear functional spaces and build, on a data selected space among
this collection, the least-squares estimator. We study the
performance of an estimator which is obtained by modifying this
least-squares estimator on a set of small probability. For the
so-defined estimator, we establish nonasymptotic risk bounds that
can be related to oracle inequalities. As a consequence of these, we
show that our estimator possesses adaptive properties in the
minimax sense over large families of Besov balls Bα,l,∞(R) with R>0, l ≥ 1 and α > α1
where α1 is a positive number satisfying
1/l - 1/2 ≤ α1 < 1/l. We also study the particular case where
the regression function is additive and then obtain an additive
estimator which converges at the same rate as it does when k=1.

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